The primary study material for AIME are the past AIME contest problems and solutions as well as the advanced AoPS series texts:
If you have qualified for AIME, you have probably studied the more elementary AopS books fully, and can expect to get 4-6 or less with just these basics.
If you want to score 7+ on the AIME, you will need to study the more advanced AoPS books listed above, as well as past AIME problems and solutions.
The official solution sets generally provide a single solution to a given problem, demonstrating the feasibility of solution within the standard high school curriculum.
However, AIME questions are often quite rich and have many possible alternate solutions, which also should also be studied for effective contest preparation.
The largest collection of alternate solutions are available on the AoPS web sites' AIME Problems and Solutions page at https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions
| AIME 7+ Topics (These topics will get you to 7+ on the AIME) |
| Topic |
Additional Practice and Reference |
- Word Problems.
- Traditional word problems - Mixture, Work, or Rate, Time, and Distance.
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- Problems in Elementary Mathematics - Lidsky (Algebra.6)
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- Linear and non-linear systems of equations.
- Equation solving, using symmetry, knowledge of basic identities, factorization, ... .
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- A Problem Book in Algebra - Krechmar (Chapter 1)
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- Logarithms and Exponents.
- Basic rules and definition, base change, in combination with equation solving and algebraic identities.
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- A Problem Book in Algebra - Krechmar (Chapter 1)
- Problems in Elementary Mathematics - Lidsky (Algebra.4)
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- Quadratics, Cubics, Polynomials and Vieta's Formulas.
- Quadratics, Cubics and higher polynomial problems involving roots, coefficients, and Vieta's formulas.
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- Problems in Higher Algebra - Faddeev (Chapter 5.3)
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- Complex Numbers (Basic).
- Basic complex number questions.
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- Problems in Higher Algebra - Faddeev (Chapter 1.1 and 1.2)
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- Arithmetic and Geometric Sequences and Series (Basic).
- Basic sequences and series questions generally involving arithmetic and geometric progressions and combinations.
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- Chapter 7 - A Problem Book in Algebra - Krechmar
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- Combinatorial Counting (Basic).
- Basic combinatorial counting questions and the Binomial Theorem.
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- Counting, 2nd Edition - Meng, Guan (2013)
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- Triangle Geometry (Basic).
- Application of the most common theorems - Pythagorean, Similar Triangles, Angle Bisector Theorem, Area, Stewart's Theorem, Cevians, Medians, ... .
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- Problems in Elementary Mathematics - Lidsky (Geometry A)
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- Triangle Geometry (Mass Point).
- Application of all the techniques - Cevians (Medians, Altitudes, Bisectors,) Transversals, Splitting, ratios and in combination with elementary triangle theorems.
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- Number Theory (Elementary).
- Modular Arithmetic, LCM, GCD, Primes, Base Arithmetic, Integer Algebra, ...
|
- Elementary Number Theory: A Problem Oriented Approach - Roberts
/ |
- Number Theory (Divisors).
- Application of the most common theorems - Number of Divisors, Sum of Divisors, Product of Divisors, "Revese" problems in each of these areas.
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- Elementary Number Theory: A Problem Oriented Approach - Roberts
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- Probability.
- Conditional, unconditional, geometric, and elementary combinatorial probabiility questions occur regularly, but not every year.
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- Challenging Mathematics Problems with Elementary Solutions I - Yaglom
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To qualify for the USAJMO or USAMO, you will generally need to score 10+ on the AIME.
Olympiad indices the last few years have been quite high, so even if you are scoring 130+ on the AMC, you will still need at least 9 or more to be sure to qualify for the Olympiad.
In order to to do this, you need to study and practice problems and solutions beyond those found in the AoPS books, such as are in the very dense books downloadable below.
| AIME 10+ Topics (Add in these to get to 10+) |
| Topic |
Additional Advanced Practice and Reference |
- Plane Geometry.
- Circle theorems, Cicumcircle, Incircle, Ptolemy's Theorem, Power of a Point, various other theorems, and in combination.
|
- Problems in Elementary Mathematics - Lidsky (Geometry.A)
- Problems in Plane Geometry - Sharygin
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- Solid Geometry.
- There is generally at least one solid geometry problem on the AIME.
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- Problems in Elementary Mathematics - Lidsky (Geometry.B)
- Problems in Solid Geometry - Sharygin
|
- Analytic Geometry.
- Equations of lines, distance of point to line, equations of circles, ellipses, parabolas, and hyperbolas, area of triangle formulas, ... .
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- Problems in Higher Algebra - Faddeev (Chapter 1.3 and 1.4)
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- Complex Numbers.
- Advanced complex number questions, roots of unity, the geometry of complex numbers in the Argand plane.
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- Problems in Higher Algebra - Faddeev (Chapter 1.3 and 1.4)
|
- Number Theory.
- Fermat's, Euler's, Wilson's Theorems, linear congruences, Chinese Remainder Theorem, Diophantine Equations, ...
|
- Elementary Number Theory: A Problem Oriented Approach - Roberts
|
- Polynomials.
- Polynomial problems involving roots and coefficients, Vieta's formulas, Newton's formulas, symmetric functions.
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- Problems in Higher Algebra - Faddeev (Chapter 5)
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- Combinatorial Counting and Probability.
- Combinatorial counting and probability questions tend to be of the advanced type, and occur about every other year.
|
- Counting, 2nd Edition - Meng, Guan (2013)
- Principles and Techniques in Combinatorics - Chen, Koh (1992)
|
- Triangle Trigonometry.
- Application of the most common theorems - Law of Cosines, Sines, Tangents, ... combined with elementary theorems, ... .
|
- Problems in Elementary Mathematics - Lidsky (Plane Geometry)
|
- Trigonometric Equations and Identities.
- Sum formulas, advanced identities, De Moivre's Theorem, , ... .
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- Problems in Elementary Mathematics - Lidsky (Trigonometry)
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- Sequences and Series.
- Recursive relation questions.
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- A Problem Book in Algebra - Krechmar (Progressions and Sums)
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- Geometric Modeling.
- Solving analytic problems with a geometric model.
|
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- Functions.
- Properties of sin, cos, tan, log, ln, domain and range.
|
|
| Recommended Problem and Review Books |
|---|
| Book | Size |
|---|
| Challenging Problems in Algebra - Posamentier,Salkind 1970 (Dover)
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| Challenging Problems in Geometry - Posamentier,Salkind 1970 (Dover)
|
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| Challenging Problems with Elementary Solutions I - Yaglom, Yaglom, 1964 (Dover)
|
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| Problems in Elementary Mathematics - Lidsky |
7.3 MB |
| Problems in Higher Algebra - Faddeev |
3.8 MB |
| A Problem Book in Algebra - Krechmar |
9.2 MB |
| Problems in Plane Geometry - Sharygin (MIR,1982).pdf |
14.5 MB |
| Problems in Solid Geometry - Sharygin (MIR,1986).pdf |
3.8 MB |
| The USSR Olympiad Problem Book - Shklasrsky, Chentzov, and Yaglom |
3.2 MB |
| Elementary Number Theory: A Problem Oriented Approach - Roberts
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| Counting, 2nd Edition - Meng, Guan (2013)
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| Principles and Techniques in Combinatorics - Chen, Koh (1992)
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| More advanced ... |
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| Mathematical Problems and Puzzles from the Polish Mathematical Olympiads |
21.4 MB |
| 250 Problems in Elementary Number Theory - Sierpinski |
6.7 MB |
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Please ask one of the coaches for additional study recommendations.